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[HM] Pappus' Commentary on Book X of Euclid's Elements.
Posted:
Dec 21, 2005 1:15 PM


Dear Julio and All,
I have been studying Pappus' Commentary on Book X of Euclid's Elements. The original of this work was of course lost and it survived only in Arabic translation. The passage I am interested in is a famous one, or based on a well known writing of Aristotle. (p.65 of William Thomson translation) Pappus is explaining the difference between continuous magnitude and discrete number and says:
"The reason for this is that numbers, progressing by degrees, advance by addition from that which is a minimum, and proceed to infinity (or indefinitely); whereas the continuous quantities begin with a definite (or determined) whole and are divisible (or subject to division) to infinity (or indefinitely) ." He goes on to say:
"If, then, the reason be demanded why a minimum but not a maximum is found in the case of a discrete quantity, whereas in the case of a continuous quantity a maximum but not a minimum is found, you should reply ...." He continues with an explanation (from Aristotle, I believe) using Pythagoreans contraries.
My problem here is : Is it even true? Or, if it is true, what does it mean to say, in the context of Euclid's Elements, that a whole of a continuous magnitude cannot be extended? Throughout Euclid one finds such as: "extend AB to C", and it is assumed this can be done indefinitely, just as it is assumed, without explicit postulation, that units can be manufactured indefinitely. The only (doubtful) example I can think of is the perimeter of a circle or surface of a sphere which could not be extended without extending the radius.
I am struggling to find an understanding of this passage in Pappus. Since this is a commentary on Euclid one would expect it to reflect Pappus' understanding of him, but it seems an artificial constraining of ideas into a Pythagorean mold, rather than a comment on Euclid. However, it was on such (mis)understandings that later mathematics was founded, not on our understanding of Euclid.
I am studying in "The commentary of Pappus on book x of Euclid's Elements; Arabic text and translation by William Thomson; with introductory remarks, notes, and glossary of technical terms by Gustav Junge and William Thomson. Cambridge, Harvard university press, 1930. Let me take the opportunity to express gratitude to William Thomson and Gustave Junge and the editors of this excellent production. My life would be meager indeed but for the work of such scholars.
Regards from cold, sunny Comer, Georgia, Bob



